3 Coordinates. 3.1 Prologue

Transcription

1 3 Coordinates 3. Prologue Coordinates describe the location of a point or an object using numbers. If ou are shipwrecked somewhere off the coast of Patagonia and radioing for help, ou can sa, I am somewhere off the coast of Patagonia. Rescuers will arrive sooner, though, if ou give a more precise location; for eample, I am 50 miles northeast of Cape Horn. It would be even better to consult our GPS (Global Positioning Sstem) device and report: I am at S W. A latitude-longitude pair serves as coordinates of a point on Earth. Coordinates connect geometr with algebra and analtical methods; computations replace measuring. When ou move a cursor on a computer screen with a mouse, the computer converts the signals received from the mouse into numbers, computes the cursor coordinates on the screen, then displas the cursor at the appropriate location. Copright 00 b Sklight Publishing Direct measuring of distances and angles is a slow, low-tech process. It ma work if ou want to mark a boundar around our field, la out the foundation for a house, or measure the distance between two points on a map with a compass and a ruler. But if ou need to launch a satellite into orbit, control an airplane on autopilot, or generate computer animations for special effects in a movie, ou need some serious computations. Knowing the coordinates of points, on the plane and in space, we can use a computer to calculate distances and directions between them. It takes onl one number to describe the location of a point on a line or a curve. For eample, a milepost or an eit number ma describe our location on a highwa. The location of a point on the number line is described b the number it represents. It takes two numbers to describe a location on a plane or a surface and three numbers to describe a location in space. If ou travel in time, it takes four numbers to describe our location in time and space. Lines are said to have one dimension; surfaces and planes, two dimensions; space, three dimensions; and space-time, four dimensions. Mathematicians and phsicists often work with multi-dimensional spaces.

2 CHAPTER 3 ~ COORDINATES 3. Cartesian Coordinates As we saw in Section <...>, there is a one-to-one correspondence between real numbers and points on the number line. Now let us consider two number lines on a plane, such that the are perpendicular to each other and intersect at zero. Let s call them and aes. B convention, the -ais is drawn horizontall, pointing to the right, and the -ais is drawn verticall, pointing up: Copright 00 b Sklight Publishing For an point P on the plane, we can draw a line through P perpendicular to the -ais. Suppose that line intersects the -ais at point M. Another line through P, perpendicular to the -ais, intersects the -ais at point N: M -3 - P N 3 The points M and N are called the projections of P to the - and -ais, respectivel. The points M and N correspond to some numbers and on the respective number lines. The ordered pair (, ) is called the Cartesian coordinates of the point P. (It is called an ordered pair because the order in which and are listed matters: the first number is the -coordinate, the second the -coordinate. We will write P(, ) to indicate that and are the coordinates of P.

3 CHAPTER 3 ~ COORDINATES 3 The - and -aes define a coordinate sstem on the plane. The intersection of the aes is called the origin of the coordinate sstem. The coordinates of the origin are (0, 0). The aes divide the plane into four regions called quadrants (Figure 3-). The are numbered in counterclockwise order. Quadrant II Quadrant I Quadrant III Quadrant IV Figure 3-. The and aes divide the plane into four quadrants Copright 00 b Sklight Publishing In the first quadrant 0 and 0. Eample -3 A C B 3 What are the coordinates of the points A, B, and C above? Solution A(-3, ), B(, ), and C(, -).

4 4 CHAPTER 3 ~ COORDINATES Eample In what quadrant does the point P(6, -3) reside? Solution Since its -coordinate is positive and its -coordinate is negative, P resides in the fourth quadrant. In a three-dimensional space, we need three numbers to describe the location of a point. In the Cartesian sstem, we take three aes,,, and z, intersecting at the origin, each perpendicular to the other two. For each point P we can take its projection to the horizontal plane -, and use the - coordinates of that projection as the and coordinates of P. The vertical position of P with respect to the - plane (positive or negative, depending on whether P is above or below the - plane), gives the third coordinate, z: Copright 00 b Sklight Publishing z P Cartesian coordinates are named after René Descartes ( ), the French philosopher, mathematician, and scientist who invented them (his Latin name was Renatus Cartesius). A Cartesian coordinate sstem establishes a one-to-one correspondence between ordered pairs of real numbers and points on the plane. In general, in an coordinate sstem, the coordinates of a point describe the point s location without ambiguit. But the converse is not required: different coordinates ma describe the same location. For eample, in the latitude-longitude coordinate sstem, the latitude 90 N corresponds to the North Pole, regardless of the longitude. Polar coordinates, eplained in Section <...>, are another eample: different coordinate pairs describe the same location on the plane.

5 CHAPTER 3 ~ COORDINATES 5 We can shift and/or rotate one or both aes in a Cartesian coordinate sstem and obtain a new Cartesian coordinate sstem. If we know what the transformation is, we can write simple equations that relate the new coordinates to the old ones. For eample, if we shift the -ais up b a positive distance b and the -ais to the right b a positive distance a P b O a O Copright 00 b Sklight Publishing the new coordinates (, ) of a point can be computed from its old coordinates (, ) as follows: = a = b These formulas work for negative a (shift to the left) and negative b (shift down), too. Eercises. Look up René Descartes biograph on the Internet. Which one of his books introduced coordinates?

6 6 CHAPTER 3 ~ COORDINATES. 3 B A D - -3 C Determine the coordinates of the points A, B, C, and D above. Which of these points is in the second quadrant? 3. Mark the points A(0, 3), B(-, -), and C(3, -) on a Cartesian coordinate grid. 4. We can describe the first quadrant as a set of all points for which 0, 0. Describe the second, third, and fourth quadrants in a similar manner. Copright 00 b Sklight Publishing 5. Name the quadrants in which the points (, -), (-, ), and (-, -) reside without drawing the points. 6. Give an eample of the coordinates of points O, A, B, and C such that O is the origin, A is not on either ais, and OABC is a square.

7 CHAPTER 3 ~ COORDINATES 7 7. TrueTpe scalable computer fonts represent each character b describing its contour. Each contour is split into line segments and quadratic splines. A spline is a segment of a smooth curve, which is inscribed into an angle whose sides are tangent to the contour: Each spline is described b three points: the two points of tangenc and the verte of the angle. The following list of 5 points defines the contour of a character: (-3, 7), (-3, 7), (-3.5, 7.5), (-, ), (3, 0), (8, 8), (3, 3), (9, 0), (4.5, -5), (, -9), (-3, -7), (-4.5, -6.5), (-3.5, -5.5), (-3, -4.5) (-.5, -5 ), (4, -9), (4, -3), (4,.5), (-,.5), (-,.5), (-, ), (5, 3), (3, 7), (, ), (-3, 7) Copright 00 b Sklight Publishing This contour consists of segments. The first point, third point, and so on are on the contour; the points in between are the vertices of spline angles. (A duplicate point indicates that the spline is actuall a straight line segment.) Sketch the character. 8. If we shift the -ais 3 units to the right and the -ais units up, what are the coordinates of P(4, -) in the new coordinate sstem? 9. If we shift the -ais 3 units to the left and the -ais units down, what are the coordinates of P(4, -) in the new coordinate sstem? 0. In computer graphics sstems, the origin of the coordinate sstem is often placed in the upper-left corner of the screen, with the -ais pointing to the right and the -ais pointing down. If the screen is 680 b 050 piels (picture elements), what are the screen coordinates of a point located one quarter of the screen to the right and three quarters of the screen down?

8 8 CHAPTER 3 ~ COORDINATES. In computer graphics, a picture within a window on the screen is often described in coordinates relative to that window (with the origin in the upperleft corner of the window and the -ais pointing down). That wa, if ou move the window on the screen, the description of the picture does not change. Computer software and/or hardware can translate the relative window coordinates into absolute screen coordinates. What are the absolute screen coordinates of a point, if its window coordinates (in piels) are (00, 50), the size of the window is 400 b 450, and the window is centered on the 680 b 050 screen?. If we rotate the coordinate aes b 90 counterclockwise around the origin, what are the coordinates of P(4, -) and Q(3, 5) in the new coordinate sstem? 3. Given three points, A (, 7), B (, ), and C (6, ), what should be the coordinates of the point D for ABCD to be a parallelogram? 4. Propose a necessar and sufficient condition for the points A(, ), B(, ), C ( 3, 3), and D ( 4, 4) to form a parallelogram. Copright 00 b Sklight Publishing 5. What are the coordinates of the midpoint of a straight line segment that connects the points A(, ) and B(, )? 6. Prove algebraicall (not using geometr) that the midpoints of the sides of an quadrilateral form a parallelogram. Hint: see Questions 4 and Distance Recall that the distance between two points on a number line is equal to the absolute value of the difference of the corresponding numbers: d = This is the length of the segment that connects the points. The same concept applies to the coordinate plane:

9 CHAPTER 3 ~ COORDINATES 9 The distance between two points on the plane is equal to the length of the line segment that connects the points. If the points A(, ) and B(, ) both lie on the -ais, or if AB is parallel to the -ais, then the distance between A and B is equal to. If A and B both lie on the -ais, or if AB is parallel to the -ais, then the distance between A and B is equal to. In the general case, we use the Pthagorean theorem to find the distance between A and B (Figure 3-): d = +. A(, ) d B (, ) Copright 00 b Sklight Publishing Figure 3-. d = + This formula works for an positions of points A and B on the plane, regardless of the signs of,,, and regardless of the relationships between these numbers, including the cases when AB is parallel to the -ais (then = ), when it is parallel to the -ais (then = ), even when A and B are the same point! Since c = c for an real number c, we can use parentheses instead of the absolute values: d = ( ) + ( ), or d = ( ) + ( ). This is called the distance formula. Eample What is the distance between A(0, -) and B(-, 3)?

10 0 CHAPTER 3 ~ COORDINATES Solution ( ) ( ) ( ) ( ) d = ( ) ( ) = + 5 = + 5 = 6. The distance from the origin of a point P(, ) is r = +. r = +. This is a special case of the distance formula, because the coordinates of the origin are (0, 0). In geometr we measured distances with a ruler; now we can use the distance formula to compute the distance between two points with known coordinates. In theor, we can now prove an geometr theorem and solve an geometr problem using onl algebra! Eample Copright 00 b Sklight Publishing Find the height of an equilateral triangle if the length of its side is a. Solution Let us solve this problem without drawing a sketch, without using geometr at all. Let ABC be an equilateral triangle: AB= BC = AC = a. Let us introduce a coordinate sstem with the origin at C and the -ais going along the line CA, so that the coordinates of C are (0, 0) and the coordinates of A are (a, 0). Suppose the coordinates of B are (, ). Then is the height of the triangle, so we need to find. We have AB= BC = a. From the distance formula, BC = a = + AB = a = ( a) + Comparing the first equation to the second we get = ( a). Recall that ( a) a a = +. So = a+ a a a = a = a =.

11 CHAPTER 3 ~ COORDINATES Plugging a a a = into the first equation we get: = + a a 3a = a = a = = a. The concept of distance is ver general: in different contets and in different mathematical theories, distance ma mean different things. If ou ask, for eample, what is the distance from Atlanta to Boston, ou probabl do not mean the length of the line segment that connects Atlanta and Boston in three-dimensional space (unless ou are going to build a tunnel from Atlanta to Boston). More likel, ou are interested in the distance along the surface of the Earth, or, if ou are driving, the travel distance along Interstate 95. On a chessboard, if ou are onl allowed to move horizontall and verticall, ou might find it useful to consider the distance between the points A(, ) and B(, ) defined as d( A, B) = +, which is different from the standard distance formula. But no matter how ou define the distance d( A, B ), a well-behaved distance must alwas have the following three properties: Copright 00 b Sklight Publishing d( A, A ) = 0 the distance from an point to itself is 0. d( A, B) = d( B, A) the distance is smmetric: for an two points A and B, the distance from A to B is equal to the distance from B to A. d( A, B) + d( B, C) d( A, C) the triangle rule: the sum of the lengths of two sides of a triangle never eceeds the length of the third side; that is, the direct route from A to C is the shortest. For the distance formula on the coordinate plane, the first two properties are obvious. At first sight, the triangle rule is obvious, too: anone knows that a straight line is the shortest distance between two points. But how do we know it, mathematicall speaking? Precisel from the triangle rule! Its proof actuall takes some work. The first step is to prove the triangle rule for the number line: for an real numbers,, and 3, Here the triangle is squashed into one dimension. The proof in this case is straightforward: we simpl consider different configurations of,, and = 3 when is between and 3, and + 3 > 3 when is outside [, 3].

12 CHAPTER 3 ~ COORDINATES The second step is to prove the triangle rule on the plane for the special case when A, B, and C lie on the same straight line. The proof follows from the previous step: just turn the line into a number line. In the final step, we prove the general triangle rule. To do that, we use the projection of B onto the line AC. Let us denote that point as P : A B P C From the triangle rule for points on the same line, AP+ PC AC. But AP AB and PC BC. So, AB+ BC AC. AB+ BC = AC if and onl if the points A, B, and C lie on the same straight line, and B lies between A and C. Copright 00 b Sklight Publishing A set of points with a well-behaved distance defined for an two points is called a metric space. Eample 3 Let us consider the set of all possible positions of the minute hand on the face of a clock and define dt (, T ) as the number of minutes it takes the minute hand to travel from the position T to the position T. Can we use dt (, T ) as the definition of distance between T and T? In other words, does dt (, T ) have the three properties of a well-behaved distance? Solution It is eas to see that dt (, T ) = 0, and that the triangle rule is satisfied, too. But dt (, T ) is not smmetric: tpicall dt (, T) dt (, T). For eample, it takes five minutes for the minute hand to travel from to 3 and 55 minutes to travel from 3 to. Therefore, dt (, T ) is not an acceptable definition for distance.

13 CHAPTER 3 ~ COORDINATES 3 Eercises. What is the distance on the coordinate plane between A(-4, ) and B(, -)?. What is the distance of P(-5, ) from the origin? 3. The vertices of a triangle have the coordinates (-, ), (0, -), and (, 3). Is this triangle equilateral, isosceles, or scalene? Find its perimeter. 4. The lengths of the bases of an isosceles trapezoid are 6 and 0; the distance between the bases is 3. Draw this trapezoid in a convenient place on the coordinate plane and find the lengths of its side and the diagonals, using the distance formula. 5. Show, using the distance formula, that A(-4, -), B(-, -), and C(5, ) lie on the same straight line. Copright 00 b Sklight Publishing 6. Using coordinates, prove that for an rectangle ABCD and an point M on the plane, inside or outside the rectangle, as well as on its border, MA + MC = MB + MD. Hint: choose a convenient origin and directions of the - and -aes. 7. What is the length of the diagonal of a cube if the length of its side is unit?

14 4 CHAPTER 3 ~ COORDINATES 8. The distance between two points in the three-dimensional space is defined as the length of the line segment that connects the points. Come up with a formula for the distance between A(,, z ) and B(,, z ). Hint: z A B z z (,, 0) (,, 0) Copright 00 b Sklight Publishing 9. Recall that a line segment that connects a verte of a triangle with the midpoint of the opposite side is called a median of the triangle. If A(, ), B(, ), and C ( 3, 3) are the vertices of a triangle, the point , is called the center of gravit of the triangle. Using onl the distance formula, show that the center of gravit of a triangle lies on each of its medians (and, therefore, all three medians intersect at the center of gravit). Hint: see Question <...> in Section <...> 0. The distance between two points on Earth s surface is measured b the length of the shortest arc that connects the points. That arc lies on the circle centered at Earth s center (assuming that Earth is a perfect sphere). When two points have the same longitude, it means the lie on the same meridian, so the distance between them is measured along the meridian. Find on the Internet the approimate radius of the Earth and the coordinates of Toronto and Miami, and estimate the distance between these two cities.. Eplain wh the direct flight route from New York to New Delhi passes close to the North Pole.. Suppose that for an points A(, ) and B(, ) on the plane we define d( A, B) = +. Show that d( A, B ) has all three properties of distance.

15 CHAPTER 3 ~ COORDINATES 5 3. Suppose that for an points A(, ) and B(, ) on the plane we define { } d( A, B) = ma, (the largest of the numbers, and ). Show that d( A, B ) has all three properties of distance. 4. Given two three-letter words, we can link them with a chain of words in which onl one letter changes from one word to the net. To go from BAT to MAN, for eample, we can use the chain BAT => CAT => CAN => MAN. Let s define the distance between two three-letter words as the length of the shortest chain that connects them (assuming that an two three-letter words can be connected b a chain). For eample, the distance from BAT to MAN is equal to, because the chain BAT => MAT => MAN eists, and no shorter chain connects these words. Does this definition of distance on the set of all three-letter words turn this set into a metric space? In other words, is this a well-behaved definition of distance? Eplain. 5. Let us define the distance between two polgons F and F on the plane as the shortest distance between a verte of F to a verte of F. Is this a well-behaved definition of distance? Copright 00 b Sklight Publishing 3.4 Relations and their Graphical Representations Mathematicians like to generalize. When the see eamples of similar situations or phenomena the tr to come up with a general abstract concept that captures the essence of the eamples. The then define the concept formall and give it a name. The stud the concept as a whole and appl it to other eamples. The concept of a relation is one such ver general abstract concept. If A and B are two sets, a relation from A to B is a set of ordered pairs (a, b), where a is an element of A and b is an element of B.

16 6 CHAPTER 3 ~ COORDINATES Eample Let L be a set of 3 letters: L = { a, b, c }. Let W be a set of four words: W = { bat, cat, can, man }. Let us consider a relation R, which is a set of all pairs (letter, word) such that letter occurs in word. R consists of seven pairs: R = {( a, bat ), ( a, cat ), ( a, can ), ( a, man ), ( b, bat ), ( c, cat ), ( c, can )}. This relation represents the relationship letter occurs in word. Note that more than one letter from L ma occur in the same word, for eample, ( a, bat ), ( b, bat ), and the same letter ma occur in more than one word from W, for eample, ( c, cat ), ( c, can )). Eample The set of all pairs of real numbers (, ) such that. = 4 is a relation from to Copright 00 b Sklight Publishing The set of all pairs (a, b), where a is an element of A and b is an element of B, is called the Cartesian product of A and B and denoted as A B. The word product is used, because when A and B are finite sets with m and n elements, respectivel, then A B has m n elements. In Eample, L contains 3 letters, W contains 4 words, and L W consists of all possible letter-word pairs. B definition, a relation is a subset of A B. The - coordinate plane can be thought of as and an set of points on the plane is a relation from to. An interesting relation tpicall represents a particular relationship between elements of A and elements of B: a pair (a, b) belongs to the relation if the relationship of interest between a and b eists. That is wh the term relation is used. Often we consider relations from a set to itself. For eample, A and B can be the same set of people.

17 CHAPTER 3 ~ COORDINATES 7 Eample 3 Let S be the set of all Facebook subscribers, R the set of all ordered pairs ( s, s ) such that s is a friend of s. This relation is smmetric: if ( s, s ) is in R then ( s, s ) is also in R. In general, a relation from a set to itself does not have to be smmetric. The relation in Eample is not smmetric. In this book we are primaril interested in relations from to. There are different was to describe such a relation: in a table, as a plot, in words. But the most common and useful wa is to use algebraic equations and/or inequalities. Eample 4 The set of all pairs (, ) such that is a relation. Eample 5 Copright 00 b Sklight Publishing The set of all pairs (, ) such that and <= is a relation. In an algebraic description of a relation, the word and is often replaced b a comma or b a curl brace. For eample, and can be written as, or This eample is a sstem of inequalities. We can have a sstem of equations, too. Sometimes we mi equations and inequalities in one sstem. As we know, in a Cartesian coordinate sstem there is a one-to-one correspondence between all points on the plane and their coordinates (, ). So an relation on the set of real numbers can be represented as a set of points on the plane. The graphical representation of a relation on the - plane is called its plot or graph.

18 8 CHAPTER 3 ~ COORDINATES Eample 6 Consider a relation between real numbers and defined b the inequalit + 4. Plot this relation on the coordinate plane. Solution In the geometric interpretation, this relation describes all points whose distance from the origin is less than or equal to. These are the points inside and on the border of the circle of radius centered at the origin: Copright 00 b Sklight Publishing An equation tpicall describes a line or several lines. An inequalit tpicall describes a region (or several disjoint regions). When a relation is described b a sstem of inequalities, its plot can be constructed as the intersection of the regions defined b the individual inequalities. It is sort of like a Venn diagram, onl instead of random blobs we use precise regions. Eample 7 Plot + 4, 0, 0.

19 CHAPTER 3 ~ COORDINATES 9 Solution Eample 8 Plot the graph of the relation defined b a sstem of inequalities Copright 00 b Sklight Publishing Solution The graph of consists of all the points on and above the parabola =. The graph of consists of all the points on and to the right of the sidewas parabola. The graph of the sstem of the two inequalities is the intersection of the two graphs. The two parabolas intersect at points (0, 0) and (, ), forming a petal : With a little knowledge of calculus it is eas to show that the area of the petal is eactl one-third of the area of the unit square into which it is inscribed. But Archimedes obtained this result without calculus, in the third centur BC. We will tell ou how he did it in Chapter <...>.

20 0 CHAPTER 3 ~ COORDINATES When a relation is described b equations connected with or, its graph can be constructed as the union of the regions defined b the individual relations. Eample 9 Plot the relation Solution =. = means = or = = - = Copright 00 b Sklight Publishing As we mentioned earlier, a relation does not have to be defined b formulas: it can be described in words, or b other means. Eample 0 Consider the set of all pairs (, ), such that and are positive integers and is evenl divisible b. Plot this relation on the coordinate plane.

21 CHAPTER 3 ~ COORDINATES Solution The plot consists of discrete points: Cartesian coordinates revolutionized mathematics because the connected geometr to algebra: the provided a wa to describe geometric figures with algebraic formulas and to visualize algebraic relations as geometric figures. Eample Copright 00 b Sklight Publishing A set of points on the - plane is described b the inequalities 5; 7. What geometric figure is it? Solution It is a rectangle with the horizontal dimension 3 and the vertical dimension 6. Eercises In Questions -6, plot the relation without using a calculator.. + = <

22 CHAPTER 3 ~ COORDINATES , 3, where the points (, ) are in the first quadrant 8. 3, 3 9. > 0. >, >.. 3. ( )( + 3) Copright 00 b Sklight Publishing = = 0 6. ( ) = + a c Give an eample of a relation from to such that its graph is one vertical line. The same for one horizontal line.

23 CHAPTER 3 ~ COORDINATES 3 8. Come up with an inequalit that describes all points on and inside the circle of radius 3 centered at (, 0). 9. A relation on a set is called smmetric if with an pair ( a, a ) it also contains the pair ( a, a ). What is the geometric interpretation of a smmetric relation from to? Which of the relations in Questions -6 are smmetric? How can we tell whether a relation described b equations and/or inequalities in, is smmetric, without plotting it? 0. Suppose a relation on real numbers with an pair (, ) also contains the pair (-, ). Such a relation is not necessaril smmetric in the sense of the definition in Question 9, but its graph does possess a kind of smmetr. What kind? Which of the relations in Questions -6 have this kind of smmetr?. Suppose a relation on real numbers with an pair (, ) also contains the pair (-, -). What kind of smmetr does a graph of such a relation posses? Which of the relations in Questions -6 have this kind of smmetr? Copright 00 b Sklight Publishing. Recall that in geometr the set of all points that satisf a given condition is called a locus of points (that satisf the condition). Given two points, A and B, find the locus of points whose distance from A is twice the distance from PA B (that is, all points P, such that PB = ). Use coordinates but interpret the result in geometric terms. Hint: Take the coordinate sstem with the -ais directed along AB with the origin at A; find the two points P and P on the -ais that belong to our locus of points; then shift the origin to the PA middle between P and P, write an equation for = in the new PB coordinates and simplif it. 3. In Question, replace with an positive number k. Draw the locus of points P such that PA k PB = for k =, k =, k =, k =, and k = 5. 5

24 4 CHAPTER 3 ~ COORDINATES 3.5 Polar Coordinates Cartesian coordinates are not the onl wa to establish a coordinate sstem on the plane. In man applications it is more convenient to use polar coordinates. For eample, if a radar screen shows an airplane approaching the airport, the air traffic controller might be interested in the airplane s distance from the airport and the direction from which the airplane is approaching. Polar coordinates use the distance of a point from the origin and the direction toward the point from the origin to describe the point s location. Copright 00 b Sklight Publishing In polar coordinates, distances are measured from one fied point, called the pole. The pole is like the origin in the Cartesian sstem. The distance from a point to the pole is called the radius and is often denoted b r (Figure 3-3). Angles are measured from one base direction, called the polar ais. B convention, the polar ais is shown in graphs as a horizontal ra, starting at the pole and pointing to the right. It is like the right half of the -ais. (Often the whole ais is drawn for convenience.) The polar ais also determines the units for measuring distances. Angles are measured in the counterclockwise direction, starting from the polar ais. The angular position of a point is often denoted b θ. The (, r θ ) pair serves as the polar coordinates of the point (Figure 3-3). pole 0 r θ Prθ (, ) 3 4 polar ais Figure 3-3. Polar coordinates In practical fields, such as navigation and surveing, and in militar applications, angles are measured in degrees. In mathematics and science, angles are often measured in radians.

25 CHAPTER 3 ~ COORDINATES = π radians. radian = π degrees. Measuring angles in radians is often more convenient for mathematical formulas and scientific computations because the radian measure corresponds to the length of the arc of the unit circle (the circle of radius centered at the origin) that spans the angle. So, if a particle is moving counterclockwise along the unit circle at a speed of unit per second, the angle θ changes at the rate of radian per second. On a Cartesian plane, points are plotted on a rectangular grid. For polar coordinates we use a polar coordinate grid, something like this: π 3 π π 3 5π 6 π 6 π Copright 00 b Sklight Publishing Eample A 7π 6 0 4π 3 D 3 3π 5π 3 π 6 B C What are the polar coordinates of points A, B, C, and D above?

26 6 CHAPTER 3 ~ COORDINATES Solution A, π 3, 7 B 3, π 6, C.5, π 6, (, 0) D. A (, r θ ) pair alwas defines a unique point on the plane. But the converse is not true: the same point can have multiple representations in polar coordinates. In fact, an point has an infinite number of representations in polar coordinates! First, an pair (0, θ ) (with radius 0 and an value of θ ) represents the pole. Second, for an point, if we add or subtract π the full revolution around the unit circle to/from θ, we get the same point. In Eample above, we gave the polar, π, or coordinates of point D as (, 0 ), but (, π ), (, 4π ), (, 00π ), ( ) (, 00π ) represent the same point. Third, the radius r can be negative! Convention permits negative radii in order to make the graphs of some polar functions consistent and continuous. Copright 00 b Sklight Publishing A negative radius r is interpreted as a positive radius with the same absolute value, but pointing in the opposite direction. This means if we flip the sign of the radius and, at the same time, add π to θ (or subtract π from θ ), we get the same point. For eample, D(, π ) represents the same point as D (, 0), and B 3, π 6 represents the same point as 7 B 3, π 6. Eample Plot on a polar grid 5 A.5, π, (, 0) B, C.5, π 3, and D( 3, 6π ).

27 CHAPTER 3 ~ COORDINATES 7 Solution π Equivalent simplified coordinates for these points are A.5, 4 C.5, π 3, and ( 3, 0) D., (, ) B π, A B 0 D 3 C Copright 00 b Sklight Publishing A polar coordinate sstem is related to a Cartesian coordinate sstem with the origin at the pole, the -ais directed along the polar ais, and the same distance units: r θ Prθ (, ) It is not ver hard to convert polar coordinates into related Cartesian coordinates. Let us first take a point Prθ (, ) in the first quadrant. Let M be its projection onto the -ais.

28 8 CHAPTER 3 ~ COORDINATES O r θ M P OPM is a right triangle with the hpotenuse OP. You might recall from geometr that in a right triangle the ratio of the length of the leg adjacent to θ to the length of the hpotenuse is called the cosine of θ and denoted cosθ ; the ratio of the length of leg opposite θ to the length of the hpotenuse is called the sine of θ (sinθ ). In OM PM other words (or rather in smbols), = cosθ and = sinθ. But OM is the OP OP -coordinate and PM is the -coordinate of P in our Cartesian sstem. OP get = cosθ and = sinθ. So r r = r. We Copright 00 b Sklight Publishing = rcos θ; = rsinθ Knowing θ, we can find cosθ and sinθ with a calculator.

29 CHAPTER 3 ~ COORDINATES 9 Eample 3: π The polar coordinates of a point M are 5,. Find its Cartesian coordinates. 7 Solution π π = 5cos, = 5sin. After setting the calculator to radian mode, we find 7 7 π π cos ; sin So ; The Cartesian coordinates of M are, approimatel, (4.505,.70). Copright 00 b Sklight Publishing What about points in other quadrants? The definitions of cosθ and sinθ are etended for all values of θ in precisel such a wa that the above polar-to-cartesian conversion formulas work. We will consider the general definitions of these trig (trigonometric) functions in Chapter <...>. Meanwhile, just remember that if ou enter an value of θ in our calculator, get its cosine and sine, and plug them into the conversion formulas, the formulas will work. The formulas will work without an change even if r is negative! Eample 4: 7π Find the Cartesian coordinates of a point if its polar coordinates are 3, 8. Solution 7π 7π (, ) = ( 3)cos, ( 3)sin. Using a calculator we find 8 8 7π 7π cos ; sin Therefore, 8 8 (, ) ( 3) ( ), ( 3) (,77,.49). The point lies in the ( ) fourth quadrant, as epected.

30 30 CHAPTER 3 ~ COORDINATES Conversion from Cartesian coordinates into related polar coordinates is prett eas, too. We know that r = +. Also cosθ =, and sinθ =. We can use either r r sinθ of these two equations to find θ, or we can use tanθ = =. A graphing cosθ calculator has commands for finding θ from a given value of a trig function b using the inverse function, such as cos, sin, or tan. The onl problem is that different values of θ can produce the same value of cos, sin, and tan, and the calculator gives ou onl one of them. If (, ) resides in the first quadrant, the calculator alwas returns the correct value of θ ; otherwise ou might need to adjust θ to match the quadrant in which (, ) resides. This requires some understanding of how the trig functions behave, so let s postpone the general case until Chapter <...>. Eample 5: Find the polar coordinates of M(4, 7). Solution Copright 00 b Sklight Publishing r = = = tanθ =. After setting the calculator to radian 4 mode, we find tan.05 (radians). Since M resides in the first quadrant, this 4 value does not need an further adjustment. The polar coordinates of M are ( 65,.05 ). A relation between r and θ (an equation, an inequalit, or a combination of several equations and/or inequalities) can be plotted in polar coordinates, just like a relation between and can be plotted in Cartesian coordinates. Man simple polar equations produce interesting curves. Eample 6: Plot r = θ for θ 0.

31 CHAPTER 3 ~ COORDINATES 3 Solution 36 Copright 00 b Sklight Publishing The curve r = aθ, where a is a positive constant, is known as the Spiral of Archimedes. In this eample, a =. The spiral starts at the pole: r = 0 when θ = 0. The spiral crosses the polar ais at r = π, r = 4π r = 6π, r = 8π, and so on. An interesting question is how the spiral behaves around the pole. B definition, θ is the angle between the radius vector (the segment directed from O to P) and the polar ais. If ou approach the pole along the spiral, θ becomes smaller and smaller, so the radius vector becomes more and more horizontal (even as it becomes shorter and shorter). At the pole, the spiral must be tangent to the polar ais. But ou ll need to zoom in reall close to see that: Google polar curves and ou will find man web sites that show famous polar curves. Man sites have animations and Java applets, which let ou change the parameters of the equation or let ou see plotting in progress. Follow the link, for eample, to Famous Curves Applet Inde. Man of these curves are defined b equations that use trigonometric functions, so we will take another look at them later, in Chapter <...>. Questions <...>-<...> in the eercises ask ou to plot a couple of simple polar regions and lines.

32 3 CHAPTER 3 ~ COORDINATES Eercises. Look up Archimedes life stor on the Internet. Wh was his tomb decorated with a sphere inscribed into a clinder?. A D B C 0 3 Determine the polar coordinates of the points A, B, C, and D above (with r > 0 and 0 θ < π ). Copright 00 b Sklight Publishing 3. On a polar coordinate grid, plot the points with the following polar π coordinates: (0, 0),,, 5π, On a polar coordinate grid, plot the points with the following polar 7π coordinates:,, 3, π, 3, π Plot the following points in polar coordinates, then convert them into Cartesian coordinates. π (a) (, 0 ) ; (b), ; (c) (, π ) ; (d) 3π,. 6. Convert the following polar coordinates into Cartesian coordinates without plotting the points. π (a), 6 ; (b) 5π.5, 8 ; (c) ( 5, 6.π ) ; (d), π 7.

33 CHAPTER 3 ~ COORDINATES Convert the following Cartesian coordinates of points in the first quadrant π into polar coordinates (with r > 0 and 0 θ ): π (a) (, 3 ) ; (b) 3,, ; (d) (, 7 ). 7 ; (c) ( ) π 8. What is greater: θ (in radians) or sinθ, for 0 θ? Hint: consider the Cartesian coordinates of (, θ ). 9. Plot on a polar grid the region defined b the sstem of inequalities π π r 3, θ Write an equation in polar coordinates that describes the circle of radius centered at the pole. Copright 00 b Sklight Publishing. Write an equation in polar coordinates that describes the vertical line through the point ( 3, 0 ). Hint: write the equation in Cartesian coordinates first, then convert it to the polar form.. Write an equation in polar coordinates that describes the vertical line through the point ( 5, π ). 3. Write an equation in polar coordinates that describes the horizontal line π through the point 4,. 4. Rewrite the equation + 4 = 0 (from Question <...>(5) in Section <...>) in polar coordinates. 5. Plot the Spiral of Archimedes for θ 0.

34 34 CHAPTER 3 ~ COORDINATES 6. Using geometr, plot on the polar coordinate plane the relation r = acosθ, where a is a positive constant. Hint: a 7. Plot the relation from Question 6, using algebra. Hint: multipl both sides b r, then convert to Cartesian coordinates, then shift the origin to a,0 and rewrite the equation in new coordinates. 8. We eplained in Eample 6 that the Spiral of Archimedes is tangent to the polar ais at the pole. The curve r = acosθ also goes through the pole (see Questions 6 and 7), but it is not tangent to the polar ais. In fact, it has a vertical tangent line. How can this be? Eplain the difference between these situations. Copright 00 b Sklight Publishing 3.6 Review Concepts, terms, methods, and formulas introduced in this chapter: Cartesian coordinates - and -aes Origin Quadrants Coordinates in the new coordinate sstem when the origin shifts to (a, b): = a = b Distance between A(, ) and B(, ) : Distance properties The triangle rule Metric space Relation Sstem of equations or inequalities Graph of a relation Polar coordinates Pole Polar ais d = ( ) + ( )

35 CHAPTER 3 ~ COORDINATES 35 Radians; 360 = π (radians) (, r θ ) = (, r θ ± π) ( r, θ ) = ( r, θ ± π) r θ cosθ = r sinθ = r r = + tanθ = From polar to Cartesian coordinates: From Cartesian to polar coordinates: = rcosθ, = rsinθ. r = +, θ = tan. Copright 00 b Sklight Publishing